The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier–Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high-frequency “pollution” when using Green’s function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply- and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.

中文翻译：

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计算Stokes算符特征值的边界积分方程方法

Stokes算子的特征值和特征函数一直是深入的分析研究的主题，并在Navier–Stokes方程的研究和仿真中得到了应用。由于斯托克斯算子是二阶的并且具有无散度约束，因此计算这些特征值和相应的特征函数是一项艰巨的任务，尤其是在复杂的几何形状和高频下。边界积分方程（BIE）框架提供了健壮且可扩展的特征值计算，这是由于（a）减小了要离散化的问题的维数，以及（b）使用格林函数表示传播时不存在高频“污染”波浪。在本文中，我们详细介绍了简单和多重连接平面域上Stokes特征值问题的BIE方法的理论论证，这需要对外部域上的振荡Stokes方程的唯一性理论进行处理。然后，使用成熟的离散化BIE的技术，我们给出数值结果，证实了本文的分析要求并证明了整体方法的效率。